Preheat an oven to 450 degrees F (230 degrees C).
Sift flour, sugar, baking powder, cream of tartar, and salt into a bowl. Use a pastry cutter to chop vegetable shortening into the flour mixture until it resembles coarse crumbs. Whisk egg and milk together in a separate bowl and slowly add milk mixture to flour mixture, stirring as you pour, until dough is moistened and well-mixed. Drop dough by spoonfuls onto an ungreased baking sheet.
Bake in the preheated oven until biscuits have risen and are golden brown, 10 to 12 minutes.
You can place dough on a floured countertop and knead more flour into dough until it no longer sticks to your fingers. Pat out to desired thickness. Cut with a biscuit cutter and transfer biscuits to an ungreased cookie sheet.
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